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Abstract

Coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in Hilbert spaces. But while coercivity estimates are well known for many variational problems arising from partial differential equations, they are still an open problem in the context of boundary integral operators arising from acoustic scattering problems, where rigorous coercivity results have so far only been established for combined integral operators on the unit circle and sphere. The fact that coercivity holds, even in these special cases, is perhaps surprising, as formulations of Helmholtz problems are generally thought to be indefinite. The main motivation for investigating coercivity in this context is that it has the potential to give error estimates for the Galerkin method which are both explicit in the wavenumber $k$ and valid regardless of the approximation space used, thus they apply to hybrid asymptotic-numerical methods recently developed for the high frequency case. One way to interpret coercivity is by considering the numerical range of the operator. The numerical range is a well established tool in spectral theory and algorithms exist to approximate the numerical range of finite dimensional matrices. We can therefore use Galerkin projections of the boundary integral operators to approximate the numerical range of the original operator. We prove convergence estimates for the numerical range of Galerkin projections of a general bounded linear operator on a Hilbert space to justify this approach. By computing the numerical range of the combined integral operator in acoustic scattering for several interesting convex, nonconvex, smooth, and polygonal domains, we numerically study coercivity estimates for varying wavenumbers. We find that coercivity holds, uniformly in the wavenumber $k$, for a wide variety of domains. Finally, we consider a trapping domain, for which there exist resonances (also called scattering poles) very close to the real line, to demonstrate that coercivity for a certain wavenumber $k$ seems to be strongly dependent on the distance to the nearest resonance.